1. **State the problem:** Evaluate the integral $$\int_1^e x^{x^x} \cdot x^x \left(\ln x + (\ln x)^2 + \frac{1}{x}\right) \, dx$$. 2. **Analyze the integrand:** The integrand is $$x^{x^x} \cdot x^x \left(\ln x + (\ln x)^2 + \frac{1}{x}\right)$$. 3. **Rewrite the integrand:** Note that $$x^{x^x} \cdot x^x = x^{x^x + x}$$. 4. **Consider the function to differentiate:** Let $$f(x) = x^{x^x}$$. 5. **Find the derivative of $$f(x)$$:** Using logarithmic differentiation: $$\ln f(x) = x^x \ln x$$ Differentiate both sides: $$\frac{f'(x)}{f(x)} = \frac{d}{dx}(x^x \ln x)$$ 6. **Differentiate $$x^x \ln x$$:** Use product rule: $$\frac{d}{dx}(x^x \ln x) = (x^x)' \ln x + x^x \frac{1}{x}$$ 7. **Find $$(x^x)'$$:** $$x^x = e^{x \ln x}$$ $$\Rightarrow (x^x)' = x^x (\ln x + 1)$$ 8. **Substitute back:** $$\frac{d}{dx}(x^x \ln x) = x^x (\ln x + 1) \ln x + x^x \frac{1}{x} = x^x \left( (\ln x)^2 + \ln x + \frac{1}{x} \right)$$ 9. **Therefore:** $$\frac{f'(x)}{f(x)} = x^x \left( (\ln x)^2 + \ln x + \frac{1}{x} \right)$$ 10. **Multiply both sides by $$f(x) = x^{x^x}$$:** $$f'(x) = x^{x^x} \cdot x^x \left( (\ln x)^2 + \ln x + \frac{1}{x} \right)$$ 11. **Notice this matches the integrand:** The integrand is exactly $$f'(x)$$. 12. **Evaluate the integral:** $$\int_1^e f'(x) \, dx = f(e) - f(1) = e^{e^e} - 1^{1^1} = e^{e^e} - 1$$ **Final answer:** $$\boxed{e^{e^e} - 1}$$